Find all $x \in \mathbb{Z}$ satisfying the congruence equation $x \equiv 1 \pmod 5$

Question

Let $x-1= 5m$ where $m\in \mathbb{Z}$

let $m= n+1$ where $n\in \mathbb{Z}$

$$x-1= 5(n+1)$$

$$x-1 = 5n+5$$

$$x = 5n+6$$

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This is what I have so far. I am not sure if this is correct or if I'm on the right track.

Answer 1

You are doing fine but actually you could have just written $x = 5m+1$ where $m\in \mathbb{Z}$.

$$\{ x \in \mathbb{Z}: x \equiv 1 \pmod 5 \} = \{ x \in \mathbb{Z}: \exists m \in \mathbb{Z}, x=5m+ 1 \}$$

While it is not wrong, letting $m=n+1$ doesn't really simplify things.